Wealth, Information Acquisition and Portfolio Choice: A Correction

Wealth, Information Acquisition and Portfolio Choice: A Correction Joel Peress INSEAD There is an error in our 2004 paper Wealth, Information Acquisition and Portfolio Choice. This note shows how to correct it by adusting the hypotheses of the model. Specifically, it assumes that agents learn about the stock s mean payoff rather than about its realization. All the conclusions of the paper remain valid. There is a mistake in the derivation of agents uncontional expected utility in Wealth, Information Acquisition and Portfolio Choice (Review of Financial Stues, 17(3 (2004, pages 879 914. Some formulas need to be corrected and some assumptions adusted, but all the conclusions of the paper remain valid. In particular, it remains the case that wealthier investors collect more information and that, as a result, they hold a larger fraction of their wealth in stocks even though their relative risk aversion is not lower. The uncontional expected utility is approximated using a Taylor series expansion, in which higher order terms are dropped. In fact, these terms cannot be neglected under the payoff structure postulated in the paper. As a result, the demand for information cannot be evaluated for arbitrary preferences. In this note, we offer a solution to the issue, which relies on a slight mofication of the stock s payoff structure and supports all the conclusions of the paper. 1. The problem The expression for the contional expected utility E (U(W 2 F stated on page 906 is correct. However, equation 12 obtained by taking its uncontional expectation is not. To see why, we rewrite E (U(W 2 F as: E (U(W 2 F = U(W 0 + U (W 0 E ( W F + 1 2 U (W 0 E ( W 2 F + o(z, (1 where W W 2 W 0 is defined at the top of page 906 as: W W 2 W 0 =r f W 0 z+pw 0 α 0 z+α W 0 ( Π P P r f z C(x z+o(z. c Published by Oxford University Press 2011. doi:10.1093/rfs/hhr070

The Review of Financial Stues / v xx n x 2011 In both formulas, o(z denotes terms of an order of magnitude smaller than z (similarly o(1 used below captures terms of an order of magnitude smaller than 1. 1 This notation allows to keep track of the quality of the approximation. The first two contional moments of W are given by: E ( W F = r f W 0 z + pw 0 α 0 z + τ(w 0 λ 2 C(x z + o(z, and E ( W 2 F = V ar ( W F + [E ( W F ] 2 = V ar ( W F + o(z = τ(w 0 2 λ 2 + o(z where λ E (πz F + 1 2 V (πz F pz r f z V (πz F is investor s Sharpe ratio defined at the bottom of page 906. Note that λ is of the order 1/2. Taking the expectation of equation 1 yields: E (U(W 2 = U(W 0 + U (W 0 E ( W (2 + 1 2 U (W 0 E ( W 2 + E (o(z. (3 This expression is not identical to equation 12 in the paper because the terms E ( W l for l > 2 cannot be neglected in the Taylor series expansion, that is E (o(z is not o(z. In fact, each term in the expansion is of order zero. For example, E ( W is of order zero though E ( W F is of order 1: E ( W = r f W 0 z + E (pw 0 α 0 z +τ(w 0 E (λ 2 C(x z + E (o(z. In this expression, while λ is of the order 1/2 so λ 2 is of the order 1, E (λ 2 is of order zero. To see why, note that the numerator of E (λ 2 contains the term E ((E (πz F 2, which in turn contains E ((πz 2 (recall from page 905 that E (πz F = a 0 z +a ξ ξ z +a S S = a 0 z +a ξ (πz μθz+a S (πz + ε. Moreover, E ((πz 2 = V ar (πz + (E (πz 2 = σπ 2 z + o(z is of order 1. Hence, both the numerator of E (λ 2 and its denominator V (πz F are of order 1. As a result, E (λ 2 is of order zero, not 1. Computing E (λ 2 explicitly leads to: E (λ 2 = V ar (λ + (E (λ 2 = V ar (λ + o(1 because (E (λ 2 is of order 1, = [h 0 (i + x ]Φ + o(1 where Φ h 0n 2 + 2in + σ 2 θ (nh 2. 1 z l is referred to as a term of order l (in z. o(z l denotes terms of an order of magnitude smaller than z l, so terms of order l + m for any strictly positive integer m. 3188

Similarly, E ( W l is of order zero (though E ( W l F is of order 1 and higher. It follows that E (U(W 2 equals a sum of an infinite number of terms, none of which can be neglected: E (U(W 2 = l=0 1 l! U (l (W 0 E ( W l where U (l denotes the l th derivative of U and l! = l (l 1 (l 2... 3 2 1. This infinite series cannot be evaluated under general utility. We point out that Appenx E which proves the convergence of the approximation and appenx F which checks its quality are correct since they deal with the demand for stocks contional on agents signals, for any arbitrary precision of information. 2. A resolution We offer a solution to the problem outlined above. In the paper, investors learn about the stock s realized payoff. Here, we assume that the stock s uncontional mean is random and that investors learn about this mean rather than the payoff s realization. Specifically, we assume that ln Π πz is normally stributed with mean bz and variance σ 2 z, where b is normally stributed with mean E(b and variance σ 2 b. Agents signals S are about b rather than about π: S = b + ε, (4 where { } ε is independent of Π, θ, b and across agents and is normally stributed with mean zero and precision x. Because the signal is about the uncontional mean of π ln Π/z, its variance is not scaled by z. The signal costs C(x z. The residual supply of shares, θ, is normally stributed with mean E(θ and variance 2 (not σ θ 2 /z as assumed in the paper. We do not impose any constraint on preferences, beyond the requirement that absolute risk aversion decreases with wealth. To account for the new payoff and signal structure, we need to define some aggregate variables. The aggregate risk tolerance, n, is unchanged. But the informativeness of the price, i, implied by aggregating invidual precisions across agents needs to be normalized by agents total precision, h(i, x h 0 (i + x, obtained by adng the precisions of the private signal, the stock price and the prior: i 1 σ 2 x h 0 (i + x τ(w 0 dg(w 0, α 0 where h 0 (i 1 σ 2 b + i 2 2. (5 We also need to define a new aggregate variable, k, as: 1 k τ(w 0 dg(w 0, α 0. (6 h 0 (i + x 3189

The Review of Financial Stues / v xx n x 2011 2.1 Stock price and portfolio composition The equilibrium stock price resembles closely that of Theorem 1, except that π, p π and σπ 2 are replaced with b, p b and σb 2 and that 1/h is replaced with k/n. Theorem 1. (price and demand for stocks Assume the scaling factor z is small. Assume the information decision has been made (i.e. i, k and x are given. There exists a log-linear rational expectations equilibrium. The equilibrium price is given by p 0 1 h ln P = pz where p + r f = p 0 (i + p b (i(b μθ, (7 h 0 (i 1 σ 2 b + i 2 σ 2 θ h(i, x h 0 (i + x h n k, ( ( E(b + i E(θ + 1 σb 2 2 2 σ 2 p b 1 1 hσb 2 and μ 1 i The optimal portfolio share of stocks for an investor with a signal of precision x (possibly equal to 0 is given by α = τ(w 1 E (πz F (p + r f z + 1 2 V (πz F + o(1 W 1 V (πz F = τ(w ( 1 1 E(b W 1 σ 2 + i E(θ h(i, x 2 + i 2 2 (b μθ + x S ( 1 + 2 σ 2 p + r f σ 2 b h(i, x (8 + o(1. (9 The proof of Theorem 1 is similar to that in the paper. We conecture that the equilibrium price is given by equations 7 to 8, solve for optimal portfolios, determine the price that clears the stock market and confirm that the conecture is valid. For the price function given in equation 7, the contional mean and variance of ln Π contional on the equilibrium price (or equivalently contional on ξ b μθ and the private signal S = b + ε are given by: V (ln Π F = V (πz F = E (V (πz b, F F + V (E (πz b, F F 3190

= E (V (πz b F + V (E (πz b F = E (σ 2 z F + V (bz F = σ 2 z + V (b F z 2 =σ 2 z + o(z, and E (ln Π F = E (πz F = E (bz F = a 0 z + a ξ ξ z + a S S z + o(z, where a 0 h E(b σ 2 b + i E(θ 2, a ξ h i 2 2, and a S h x. Note that the variance of returns equals a constant σ 2 z at the order 1 since V (bz F = z 2 /h = o(z is of order 2. We turn to the determination of the optimal portfolio share. The approximate Euler equation at the top of page 906 holds: U (W 0 E [ Π P P + o(z = 0, r f z F ] + U (W 0 α W 0 E [( Π P P r f z 2 F ] [ with E Π P P r f ] z F = E (πz F + 1 2 V (πz F pz r f z+o(z = E (bz F + 1 2 σ 2 z pz r f [ z+o(z and E ( Π P P r f z 2 ] F = V (πz F = σ 2 z + o(z. Solving for the optimal share yields: α = τ(w 0 W 0 E (bz F + 1 2 σ 2 z pz r f z σ 2 + o(1 z = τ(w 0 W 0 1 σ 2 [E (b F + 1 2 σ 2 p r f ] + o(1, (10 which leads to equation 9 above. Aggregating this equation over all investors yields the aggregate demand for the stock: W 0 α P [( = 1 E(b σ 2 σb 2 + i E(θ 2 + i 2 2 (b μθ ] τ(w 0 + σ 2 ib (p + r f 1 h 2 σ 2 n + o(1, where the term πi comes from applying the law of large numbers to the sequence {τ x ε /h }. Equating aggregate demand to aggregate supply θ yields the equilibrium price given by equations 7 8. 3191

The Review of Financial Stues / v xx n x 2011 2.2 Demand for information We determine the optimal precision chosen by an agent. Theorem 2. (demand for information Assume the scaling factor z is small. There exists a wealth threshold W0 (i such that only agents with initial wealth above W0 (i acquire information. Their optimal precision level, x = x(w 0, is characterized by the firstorder contion C (x = 1 2 τ(w 0 ϕ (x ; i, (11 where ϕ measures the expected square Sharpe ratio on an investor s portfolio, an increasing and concave function of the precision of her private signal x. The function ϕ(x ; i, k, nz E (λ 2 measures the square Sharpe ratio which an investor expects in the planning period given that she will receive some information in the trang period. Specifically, λ E (πz F + 1 2 V (πz F pz r f z V (πz F = E (bz F + 1 2 σ 2 z pz r f z σ 2 z = 1 ( E (b F + 1 z. σ 2 σ 2 p r f λ is of the order 1/2. Integrating over the stributions of S and P yields: (12 E (λ 2 ϕ(x 1 ; i, k, nz = σ 2 h(i; x z + Bz [ where B 1 σ 2 ( ] θ σ 2 i 2 p2 b + σ b 2 (1 p b 2 + E(θ2 k 2 i 2 1 h 0.(13 n In contrast to the specification of the paper scussed in the previous section, E (λ 2 is of order 1, not zero. This is because E ((E (πz F 2 = E ((E (bz F 2 contains terms of the type E ((bz 2 which are of order 2, rather than terms of the type E ((πz 2, of order 1. The value of information and its cost are both of order 1. Equation 12 in the paper is valid under the new 3192

specification because the terms E ( W l for l > 2 can be neglected in the Taylor series expansion, that is E (o(z = o(z: [ 1 E (U(W 2 = U(W 0 + U (W 0 z 2 τ(w 0 ϕ(x ; i + r f W 0 ] + E (pw 0 α 0 C(x + o(z. For an informed investor, ϕ(x ; i is an increasing function of x. Moreover, in contrast to the specification in the paper, it is concave in x. This simplifies the proof of the existence and unicity of an interior optimum. Its derivative, ϕ (x ; i = 1 σ 2 (h 0 (i+x 2, is decreasing in x so the second-order contion is satisfied. This leads to the first-order contion for the precision (B does not depend on x ; ϕ and therefore x depends on i not k: C (x = τ(w 0 2σ 2 ( h 0 (i + x 2 ϕ (x ; i. This equation admits a unique solution if and only if C (0 < τ(w 0 2σ 2 h 0 (i 2 ϕ (0; i because its left hand side is monotonically increasing in x starting from C (0, while its right hand side is monotonically decreasing towards zero. This implies that agents acquire information if and only if their initial wealth is above W0 (i τ 1 ( 2σ 2 h 0 (i 2 C (0. The existence and unicity of the equilibrium (within the class of log-linear equilibria can be established for any parameter value (even when σ 2 > 2 or σb 2 > 2 unlike in the paper as stated in Theorem 3 below. Theorem 3. (equilibrium level of information and unicity Assume the scaling factor z is small. In equilibrium, price informativeness i solves i = W0 x(w 0 ; i W0 (i h 0 (i + x(w 0 ; i τ(w 0 dg(w 0, α 0. (14 There exist a unique log-linear equilibrium. We proceed as in the paper to show that the aggregate variable i is uniquely defined. Let Σ(i i W 0 x(w 0 ;i W0 (i h 0 (i+x(w 0 ;i τ(w 0 dg(w 0. Differentiating 3193

The Review of Financial Stues / v xx n x 2011 Σ yields: Σ (i = 1 + x(w 0 ; iτ(w 0 g(w 0 dw0 h 0 (i + x(w0 ; i ( W0 W0 = 1 W0 W 0 d d x(w 0 ; i h 0 (i + x(w 0 ; i ( x(w 0 ; i h 0 (i + x(w 0 ; i τ(w 0 dg(w 0 τ(w 0 dg(w 0 because x(w0 = 0 (C is continuous at 0. Differentiating the first-order contion for the precision yields dx = ( 4i/ 2 2+C (x/c (x < 0 and therefore d x 1 dx h 0 +x = 2 ix < 0. (h 0 +x 2 ( Hence, Σ (i is positive, Σ is monotonic and i (and μ is uniquely defined (Σ(0 0 because x(w 0 ; 0 0 and Σ( 0 because x(w 0 ; 0 = 0. k is uniquely and explicitly defined by equation 5 (x depends on i not k. 2.3 Wealth and portfolio shares We turn to the share of wealth invested in equity. The paper focuses on the uncontional average portfolio share, E(α. However, a better measure of equity holng is the uncontional average absolute portfolio share, E ( α. Indeed, there are no short sales constraints in the model, so agents go long the stock at times, and short at others. As a result, E(α does not capture their real equity exposure. For example, an agent who is long one share 50% of the time and short one share 50% of the time, has an E(α of zero though she is actually permanently invested in the stock market. As a matter of fact, this agent would qualify empirically as an equity holder. E( α does not suffer from this limitation, so we use it here to quantify portfolio holngs. Combining equations 10 and 12 leads to α = τ(w 0 1 W 0 σ z λ + o(1 which is approximately normally stributed so (e.g. He and Wang (1995: E( ( α τ(w0 2 = 2V ar (α /π + o(1 = 2 V ar (λ /(πσ 2 z+o(1 = τ(w 0 W 0 σ W 0 2ϕ(x ; i/π + o(1 = τ(w 0 W 0 σ σ 2 θ 2ϕ(x ; i/π + o(1. This expression shows that E( α increases with x, which in turn increases with W 0 so E( α increases with W 0 : wealthier investors invest a larger fraction of their wealth in stocks (through long or short positions. This conclusion only requires absolute risk aversion to decrease with wealth, and holds for example under CRRA utility. 3194

2.4 Portfolio returns and Sharpe ratios As in the paper, we denote r p z the net return on investor s portfolio (before accounting for the information cost and r pe z r p z r f z be the associated excess return. Using equation 9 and integrating over all the random variables yields the same expressions as in the paper (equations 11, E(r pe z = τ(w 0 ϕ(x z, V (r pe z = W 0 ( τ(w0 W 0 2 ϕ(x z, and E(r pe z = ϕ(x V (r pe z z These equations imply that wealthier agents, because they acquire more information, achieve a higher expected return, a higher variance and a higher Sharpe ratio on their portfolio. As a consequence, their average terminal wealth and certainty equivalent are greater. In that sense, the stock market magnifies inequalities. Moreover, as suggested in the paper, these equations imply that one can empirically stinguish between the information view of portfolio choice and the decreasing relative risk aversion view (which both imply that wealthier households invest a larger fraction of their wealth in equity, and achieve a higher expected return and variance on their portfolio by analyzing how portfolios Sharpe ratios vary with wealth. They increase with wealth accorng to the information view (as long as absolute risk aversion is decreasing, but are independent of wealth accorng to the decreasing relative risk aversion view. References He, H., and J. Wang. 1995. Differential Information and Dynamic Behavior of Stock Trang Volume. Review of Financial Stues 8:919 72. Peress, J. 2004. Wealth, Information Acquisition and Portfolio Choice. Review of Financial Stues 17: 879 914. 3195